We consider combinatorial exchanges where agents have (possibly random) endowments and ordinal preferences over bundles of indivisible goods. For any market instance, we show that there exists an approximately feasible, individually rational, and ordinally efficient lottery assignment. This assignment can be supported by prices derived from a novel competitive equilibrium concept, which we term a Budget-Relaxed Approximate Competitive Equilibrium (BRACE). Any BRACE can be implemented as a lottery over deterministic allocations that are approximately feasible, individually rational and efficient. When endowments are deterministic, it can be implemented over near-feasible weak core outcomes. Moreover, BRACEs are ordinally envy-free and ex-post envy-free up to one good (where envy is only justified if another agent's endowment is either smaller or worth less in equilibrium). A mechanism that implements a BRACE is strategyproof in the large. Our framework can be used in many real-world market design applications, such as organ exchanges, tuition exchanges, time bank sharing, shift exchanges, and resource reallocation.

In today's financial landscape, traditional exchanges compete against online trading platforms. A critical point of competition centers around transaction costs. While traditional exchanges adhere to transparent transaction cost structures, many online trading platforms, under the guise of 'zero-commission trading,' conceal transaction costs within their bid-ask spread. In this paper, I show that hidden transaction costs induce additional volatility in the form of price cycles in markets that would be stable if transparent transaction costs were charged. To compete with the profit opportunities from price cycles on platforms with hidden transaction costs, platforms with transparent costs must reduce them below the optimal monopolist level to attract traders. In this duopoly, I show that there is a market equilibrium: more risk-averse traders prefer transparent transaction costs, while less risk-averse traders choose hidden costs. Depending on the risk attitudes of traders, transparent transaction costs can be more or less efficient than hidden transaction costs. Finally, I demonstrate that the price cycles in markets with hidden transaction costs can be exploited through strategic market entry to a point where the platform ceases to make any revenue, highlighting the necessity of uninformed traders for sustaining zero-commission models.

Transaction costs are omnipresent in markets but are often omitted in economic models. We show that the presence of transaction costs can fundamentally alter incentive and welfare properties of Double Auctions, a canonical market organization. We further show that transaction costs can be categorized into two types. Double Auctions with homogeneous transaction costs---a category that includes fixed fees and price based fees---preserve the key advantages of Double Auctions without transaction costs: markets with homogeneous transaction costs are asymptotically strategyproof, and there is no efficiency-loss due to strategic behavior. In contrast, double auctions with heterogeneous transaction costs---such as spread fees---lead to complex strategic behavior (price guessing) and may result in severe market failures. Allowing for aggregate uncertainty, we extend these insights to market organizations other than Double Auctions.